Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 11/26/2024 2:15 PM, WM wrote:

On 26.11.2024 19:49, Jim Burns wrote:

There are no last end.segments of ℕᶠⁱⁿ
There are no finitely.sized end segments of ℕᶠⁱⁿ
There are no finite cardinals common to
 each end.segment of ℕᶠⁱⁿ

>
That is a contradiction.

It contradicts ℕᶠⁱⁿ being finite, nothing else.

If there are no common numbers,
then all numbers must have been lost.
But then no numbers are remaining.

Yes.
Each finite.cardinal k is countable.past to
  k+1 which indexes
   Eᶠⁱⁿ(k+1) which doesn't hold
    k which is not common to
     all end segments.
Each finite.cardinal k is not.in
  the intersection of all end segments,
  the set of elements common to all end.segments,
   which is empty.
No numbers are remaining.

Then there are finite endsegments because
∀k ∈ ℕ: |E(k+1)| = |E(k)| - 1.

For each cardinal.which.can.change.by.1 j
|Eᶠⁱⁿ(k)| is larger than j
|Eᶠⁱⁿ(k)| isn't j
|Eᶠⁱⁿ(k)| isn't any cardinal.which.can.change.by.1
|Eᶠⁱⁿ(k)| cannot change by 1
|Eᶠⁱⁿ(k+1)| = |Eᶠⁱⁿ(k)|